Which of the following are true and which false? Give proofs or counterexamples.I said this was false and my counterexample was .i). If (this is a mapping, I couldn't quite find the right arrow) is continous at c then f is differentiable at c.

This function is continous at 0 but which is not differentiable at 0.

I'm not really sure about this one.ii). If a,b) \rightarrow \mathbb{R}" alt="\exists \ ga,b) \rightarrow \mathbb{R}" /> and a,b) \rightarrow \mathbb{R}" alt="Za,b) \rightarrow \mathbb{R}" /> such that then f is differentiable at c if:

a). Z is bounded

b). g is differentiable at c with g'(c)=0 and

c). g(c)=0;

I think it's true but I don't know how to prove it.

I put false and used the counterexample when c=0.iii). In ii) f is differentiable assuming only (a) and (c).

and .

Once again I put false and used the counterexample where c=0.In ii). f is differentiable assuming only (a) and (b).

Where and

This is false because could be discontinous.In (ii). f is differentiable assuming only (b) and (c).

My counterexample was at c=0.

Here and .

are these correct?